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Monday, August 15, 2011

Phase Space

This is a neat bit of work and
this is worth taking your time over. Their work adds a new viewpoint to the
idea that we all hold in common about space time.It does nothing to unify that I can see, but allows
a different formulation of the equations that could be very beneficial for
empirical work.

The information paradox referred
to is a misnomer in my opinion based on the failure to recognize that particles
are converted into photons and that is how the information is carried off.

In the meantime, we have an
improvement on the available options for considering space time.

IT WASN'T so long ago we thought space and time were the absolute and
unchanging scaffolding of the universe. Then along came Albert Einstein, who
showed that different observers can disagree about the length of objects and
the timing of events. His theory of relativity unified space and time into a
single entity - space-time. It meant the way we thought about the fabric of
reality would never be the same again. "Henceforth space by itself, and
time by itself, are doomed to fade into mere shadows," declared mathematician
Hermann Minkowski. "Only a kind of union of the two will preserve an
independent reality."

But did Einstein's revolution go far enough? Physicist Lee Smolin at the
Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada, doesn't think so. He and a
trio of colleagues are aiming to take relativity to a whole new level, and they
have space-time in their sights. They say we need to forget about the home
Einstein invented for us: we live instead in a place called phase space.

If this radical claim is true, it could solve a troubling paradox about
black holes that has stumped physicists for decades. What's more, it could set
them on the path towards their heart's desire: a "theory of everything" that will finally unite
general relativity and quantum mechanics.

So what is phase space? It is a curious eight-dimensional world that merges
our familiar four dimensions of space and time and a four-dimensional world
called momentum space.

Momentum space isn't as alien as it first sounds. When you look at the
world around you, says Smolin, you don't ever observe space or time - instead you
see energy and momentum. When you look at your watch, for example, photons
bounce off a surface and land on your retina. By detecting the energy and
momentum of the photons, your brain reconstructs events in space and time.

The same is true of physics experiments. Inside particle smashers,
physicists measure the energy and momentum of particles as they speed toward
one another and collide, and the energy and momentum of the debris that comes
flying out. Likewise, telescopes measure the energy and momentum of photons
streaming in from the far reaches of the universe. "If you go by what we
observe, we don't live in space-time," Smolin says. "We live in
momentum space."

And just as space-time can be pictured as a coordinate system with time
on one axis and space - its three dimensions condensed to one - on the other
axis, the same is true of momentum space. In this case energy is on one axis
and momentum - which, like space, has three components - is on the
other (see diagram).

Simple mathematical transformations exist to translate measurements in
this momentum space into measurements in space-time, and the common wisdom is
that momentum space is a mere mathematical tool. After all, Einstein showed
that space-time is reality's true arena, in which the dramas of the cosmos are
played out.

Smolin and his colleagues aren't the first to wonder whether that is
the full story. As far back as 1938, the German physicist Max Born noticed
that several pivotal equations in quantum mechanics remain the same whether expressed
in space-time coordinates or in momentum space coordinates. He wondered whether
it might be possible to use this connection to unite the seemingly incompatible
theories of general relativity, which deals with space-time, and quantum
mechanics, whose particles have momentum and energy. Maybe it could provide
the key to the long-sought theory of quantum gravity.

Born's idea that space-time and momentum space should be
interchangeable - a theory now known as "Born reciprocity" - had a
remarkable consequence: if space-time can be curved by the masses of stars and
galaxies, as Einstein's theory showed, then it should be possible to curve
momentum space too.

At the time it was not clear what kind of physical entity might curve
momentum space, and the mathematics necessary to make such an idea work hadn't
even been invented. So Born never fulfilled his dream of putting space-time and
momentum space on an equal footing.

The quartet took the standard mathematical rules for translating
between momentum space and space-time and applied them to a curved momentum
space. What they discovered is shocking: observers living in a curved momentum
space will no longer agree on measurements made in a unified space-time. That
goes entirely against the grain of Einstein's relativity. He had shown that
while space and time were relative, space-time was the same for everyone. For
observers in a curved momentum space, however, even space-time is
relative (see diagram).

This mismatch between one observer's space-time measurements and
another's grows with distance or over time, which means that while space-time
in your immediate vicinity will always be sharply defined, objects and events
in the far distance become fuzzier. "The further away you are and the more
energy is involved, the larger the event seems to spread out in
space-time," says Smolin.

For instance, if you are 10 billion light years from a supernova and
the energy of its light is about 10 gigaelectronvolts, then your measurement of
its location in space-time would differ from a local observer's by a light
second. That may not sound like much, but it amounts to 300,000 kilometres.
Neither of you would be wrong - it's just that locations in space-time are
relative, a phenomenon the researchers have dubbed "relative locality".

Relative locality would deal a huge blow to our picture of reality. If
space-time is no longer an invariant backdrop of the universe on which all
observers can agree, in what sense can it be considered the true fabric of
reality?

That is a question still to be wrestled with, but relative locality has
its benefits, too. For one thing, it could shed light on a stubborn puzzle
known as the black
hole information-loss paradox. In the 1970s, Stephen Hawking discovered
that black holes radiate away their mass, eventually evaporating and
disappearing altogether. That posed an intriguing question: what happens to all
the stuff that fell into the black hole in the first place?

Relativity prevents anything that falls into a black hole from
escaping, because it would have to travel faster than light to do so - a cosmic
speed limit that is strictly enforced. But quantum mechanics enforces its own
strict law: things, or more precisely the information that they contain, cannot
simply vanish from reality. Black hole evaporation put physicists between a
rock and a hard place.

According to Smolin, relative locality saves the day. Let's say you
were patient enough to wait around while a black hole evaporated, a process
that could take billions of years. Once it had vanished, you could ask what
happened to, say, an elephant that once succumbed to its gravitational grip.
But as you look back to the time at which you thought the elephant had fallen
in, you would find that locations in space-time had grown so fuzzy and
uncertain that there would be no way to tell whether the elephant actually fell
into the black hole or narrowly missed it. The information-loss paradox
dissolves.

Big questions still remain. For instance, how can we know if momentum
space is really curved? To find the answer, the team has proposed several
experiments.

One idea is to look at light arriving at the Earth from distant
gamma-ray bursts. If momentum space is curved in a particular way that
mathematicians refer to as "non-metric", then a high-energy photon in
the gamma-ray burst should arrive at our telescope a little later than a
lower-energy photon from the same burst, despite the two being emitted at the
same time.

Just that phenomenon has already been seen, starting with some unusual
observations made by a telescope in the Canary Islands
in 2005 (New
Scientist, 15 August 2009, p 29). The effect has since
been confirmed by NASA's Fermi gamma-ray space telescope, which has been collecting
light from cosmic explosions since it launched in 2008. "The Fermi data
show that it is an undeniable experimental fact that there is a correlation
between arrival time and energy - high-energy photons arrive later than
low-energy photons," says Amelino-Camelia.

Still, he is not popping the champagne just yet. It is not clear
whether the observed delays are true signatures of curved momentum space, or
whether they are down to "unknown properties of the explosions
themselves", as Amelino-Camelia puts it. Calculations of gamma-ray bursts
idealise the explosions as instantaneous, but in reality they last for several
seconds. While there is no obvious reason to think so, it is possible that the
bursts occur in such a way that they emit lower-energy photons a second or two
before higher-energy photons, which would account for the observed delays.

In order to disentangle the properties of the explosions from
properties of relative locality, we need a large sample of gamma-ray bursts
taking place at various known distances (arxiv.org/abs/1103.5626).
If the delay is a property of the explosion, its length will not depend on how
far away the burst is from our telescope; if it is a sign of relative locality,
it will. Amelino-Camelia and the rest of Smolin's team are now anxiously
awaiting more data from Fermi.

The questions don't end there, however. Even if Fermi's observations
confirm that momentum space is curved, they still won't tell us what is doing
the curving. In general relativity, it is momentum and energy in the form of
mass that warp space-time. In a world in which momentum space is fundamental,
could space and time somehow be responsible for curving momentum space?

Work by Shahn Majid, a mathematical physicist at Queen Mary
University of London, might hold some clues. In the 1990s, he showed that
curved momentum space is equivalent to what's known as a noncommutative
space-time. In familiar space-time, coordinates commute - that is, if we want
to reach the point with coordinates (x,y), it doesn't matter whether we
take xsteps to the right and then y steps forward, or if we
travel y steps forward followed by x steps to the right.
But mathematicians can construct space-times in which this order no longer
holds, leaving space-time with an inherent fuzziness.

In a sense, such fuzziness is exactly what you might expect once
quantum effects take hold. What makes quantum mechanics different from ordinary
mechanics is Heisenberg's uncertainty principle: when you fix a particle's
momentum - by measuring it, for example - then its position becomes completely
uncertain, and vice versa. The order in which you measure position and momentum
determines their values; in other words, these properties do not commute. This,
Majid says, implies that curved momentum space is just quantum space-time in
another guise.

What's more, Majid suspects that this relationship between curvature
and quantum uncertainty works two ways: the curvature of space-time - a
manifestation of gravity in Einstein's relativity - implies that momentum space
is also quantum. Smolin and colleagues' model does not yet include gravity, but
once it does, Majid says, observers will not agree on measurements in momentum
space either. So if both space-time and momentum space are relative, where does
objective reality lie? What is the true fabric of reality?

Smolin's hunch is that we will find ourselves in a place where
space-time and momentum space meet: an eight-dimensional phase space that
represents all possible values of position, time, energy and momentum. In
relativity, what one observer views as space, another views as time and vice
versa, because ultimately they are two sides of a single coin - a unified
space-time. Likewise, in Smolin's picture of quantum gravity, what one observer
sees as space-time another sees as momentum space, and the two are unified in a
higher-dimensional phase space that is absolute and invariant to all observers.
With relativity bumped up another level, it will be goodbye to both space-time
and momentum space, and hello phase space.

"It has been obvious for a long time that the separation between
space-time and energy-momentum is misleading when dealing with quantum
gravity," says physicist João
Magueijo of Imperial College London. In ordinary physics, it is easy enough
to treat space-time and momentum space as separate things, he explains,
"but quantum gravity may require their complete entanglement". Once
we figure out how the puzzle pieces of space-time and momentum space fit
together, Born's dream will finally be realised and the true scaffolding of
reality will be revealed.

Bibliography

The principle of relative locality by Giovanni Amelino-Camelia and
others

1 comment:

This "Momentum Space" theory seems to be relevant to the "Wormhole Theory" and "Quantum Tunneling". Is it?

Also, it is stated the speed of light is never exceeded and "stricly enforced." This is easily proven wrong by Einstein's famous equation, E=M2. The nuclear industry is built on this: Energy equals matter traveling at the speed of light squared. The speed of light squared is much faster than the speed of light, we just don't understand how to do it beyond the quantum subatomic level with human tools.

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Apr 2017 - 4.1 Mil Pg Views, March 2013 - Posted my paper introducing CLOUD COSMOLOGY & NEUTRAL NEUTRINO rigorously described as the SPACE TIME PENDULUM, September 2010 I am pleased to report that my essay titled A NEW METRIC WITH APPLICATIONS TO PHYSICS AND SOLVING CERTAIN HIGHER ORDERED DIFFERENTIAL EQUATIONS' has been published in Physics Essays(AIP) and appeared in their June 2010 quarterly. 40 years ago I took an honors degree in applied mathematics from the University of Waterloo. My interest was Relativity and my last year there saw me complete a 900 level course under Hanno Rund on his work in relativity,as well as differential geometry(pure math) and of course analysis. I continued researching new ideas and knowledge since that time and I have prepared a book for publication titled Paradigms Shift&. I maintain my blog as a day book and research tool to retain data and record impressions and interpretations on material read. Do join my blog and receive Four items of interest daily Monday through Saturday.